Unformatted text preview: MATH 131B
2ND PRACTICE MIDTERM
Problem 1. State the book’s deﬁnition of:
(a) A complete metric space
(b)
and
(c) Convergence of a series of real numbers
(d) Normed vector space; Banach space
Problem 2. Let be a metric space with a metric . Let
and be two Cauchy sequences in
. Show that
exists. Note: we do not assume that is complete.
is closed if
Problem 3. Let be the usual Eucledian metric on . We say that a subset
whenever
and
, then
. Show that a subset
is complete with
respect to if and only if it is closed.
Problem 4. Let
be given by
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X a` Y
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U 3 ) & Show that there is a unique point
with the property that
.
Problem 5. State and prove that Banach contraction principle.
Problem 6. Let
and
be norms on the space
of continuous functions on the
, given by:
interval
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U 3r ) r & 5
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9 3r)r d ) Y
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y D
"Qy #y D
Qy d)Y
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3 &
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Qy
3 &
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y D
r f f U
ed f ¡
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f 1 if and only if Show that the two norms are not equivalent.
Problem 7. Let
and
. Show that
and moreover that if this is the case, then
.
Problem 8. State and prove the comparison test. converges, @
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ed ...
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 Winter '08
 hitrik
 Real Numbers, Vector Space, Metric space, normed vector space, complete metric space, banach contraction principle

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